where \(\gamma \)  is a hyperparameter that balances two terms in (9), and \(E(T^{\prime })=-\sum _{ij}^{mn}T^{\prime }_{ij}(log (T^{\prime }_{ij}-1)\)  is the entropy of the solution matrix \(T^{\prime }\) . It has been shown that if \(T^{\prime }_{\gamma }\)  is the solution of the optimization (9), then \(\exists ! u\in \mathbb {R}_+^n\) , \(v\in \mathbb {R}_+^m\)  such that the solution matrix for (9) is \(T^{\prime }_{\gamma }=\text{diag}(u)K\text{diag}(v)\)  where, \(K=exp(-X/\gamma )\)  [1]}. The vectors \(u\)  and \(v\)  are updated iteratively between step 1 and 2 by using the well-known Sinkhorn algorithm as follows: step \(1) u=a/{Kv}\)  and step \(2) v=b/{K^{\top }u}\) , where\(/\)  denotes element-wise division operator [1]}.
